완효성 비료와 4종복비의 혼용 시용이 팔레놉시스의 생육에 미치는 효과

Théorie de Hodge et géomětrie algěbrique complexe

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Classical Algebraic Geometry

Preface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

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Chow Rings, Decomposition of the Diagonal, and the Topology of Families

Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.

Positivity of twisted relative pluricanonical bundles and their direct images

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

Algebraic integrability of foliations with numerically trivial canonical bundle

Let \(X\) be a compact Kahler threefold that is not uniruled. We prove that \(X\) has a minimal model.

Minimal models for Kähler threefolds

These are extended notes of the talks I gave at the workshop “Rencontre positivité” in Rennes. The aim of these talks was to illustrate the interaction between the geometry of a fibration and the positivity of direct image sheaves. We give numerous examples and an introduction to the KollárViehweg method for proving direct image theorems. As an application, we study the geometry of projective manifolds with nef anticanonical bundle.

https://www.ems-ph.org/journals/show_pdf.php?issn=0013-8584&vol=56&iss=1&rank=4

Positivity of direct image sheaves a geometric point of view

Let \(X\) be a normal variety such that \(K_X\) is \(\mathbb {Q}\)-Cartier, and let \(f:X \rightarrow X\) be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism \(f\). As a consequence we prove that if \(X\) is projective and \(f\) polarised, then \(X\) has at most log-canonical singularities.

Singularities of varieties admitting an endomorphism

Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.

Andreas Höring

Papers

Algebraic integrability of foliations with numerically trivial canonical bundle

Minimal models for Kähler threefolds

Positivity of direct image sheaves a geometric point of view

Singularities of varieties admitting an endomorphism

A decomposition theorem for projective manifolds with nef anticanonical bundle